Problem: Determine how many solutions exist for the system of equations. ${-6x+2y = -14}$ ${y = 2x-6}$
Answer: Convert both equations to slope-intercept form: ${-6x+2y = -14}$ $-6x{+6x} + 2y = -14{+6x}$ $2y = -14+6x$ $y = -7+3x$ ${y = 3x-7}$ ${y = 2x-6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x-7}$ ${y = 2x-6}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.